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Objectives
• Uncertainty analysis
• Quality control
• Regression analysis
Example: Convective heat transfercalculation
h = q/(Tsurface – Tair)
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_
_
T
Tδ
q
qδhh δ
convectivesurface
convectivesurface
22airsurface TTT
…….
0 1 2 3 4 5 6 7 8 9 10 11 12 13 140.0
0.5
1.0
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0.0
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Measured in present study
Awbi and Hatton (1999)
h = 1.823 T.0.293/H0.121 (SI)
h ro
om [W
/m2K
]
Troom
[C]
h = 1.74 T.33 (SI)
hro
om [B
TU
/h ft
2°F
]
Troom
[F]
2_2
_2
__ losswallradiativesurfacetotalsurfaceconvectivesurface qqqq
How about measurement variationor time or spatial averaging ?
• Sometimes quantities of interest fluctuate• Sometimes we are interested in slopes of
lines/regression equations• Often appropriate to use standard deviation or
standard error• Be careful and acknowledge when you are doing so
Example of uncertainty reporting
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Repetition of experiments
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General procedure
• Expend substantial effort on uncertainty• Consider all sources of error
• State best guess of reasonable “best guess”
• Always say something about uncertainty
• Clearly state your procedure in your Methodology
Experimental Design
• Huge topic (we will talk about this during whole course)
• Questions:• What do you need to know?• How well (i.e. with what confidence) do you need
to know it?• How much do you expect it to vary?• What factors are likely to influence/confound your
results?
• Always find fundamental parameters
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Some Approaches
• Bench-scale experiments in laboratory, followed by large-scale laboratory tests, followed by field tests
• Field tests followed by controlled laboratory investigation
• Larger field studies
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Example 1
• What is the average efficiency of AC units in Austin (average COP) ?
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Quality Assurance (QA)Quality Control (QC)
How to incorporated QA/QC into your experimental study?
Experiment Design Phase:
• Define objective - What question are you trying to answer?
- How will you know you are finished?
• Choose - Factors of interest
- Parameters to measure
- Experiments control method(s)
- The data analysis techniques
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How to incorporated QA/QC intoyour experimental study?
Experiment Design Phase:
For measured parametersconsider:- Range- Number of points- Number of repetitions
Create an experimental matrix
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How to incorporated QA/QC intoyour experimental study?
Experimental matrix
Be real:
- Consider available time and funding
- Predict potential for failure - predict more experiments than minimum
- predict extra time for repetition
- Preliminary experiments help
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How to incorporated QA/QC intoyour experimental study?
Measuring Phase:- Use measuring techniques that will meet the needs of your experiment
- Collect sufficient data (including repetition) to adequately characterize the measured parameter
- Record all available conditions/parameters (even those that are not in your matrix)
- Use experiment control methods
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How to incorporated QA/QC intoyour experimental study?
Data Analysis Phase:
• Graphs & descriptive statistics first• Hypothesis testing • Regression next• Interpret the results• Draw conclusions
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How to incorporated QA/QC intoyour experimental study?
Be ready to modify and/or go back and forth between phases
Example…..
Example from convection correlation development study
• Repetition and multiple measurements of important variables
• Repetition of experiments
• Independent development of data processing templates
• Comparison with previous research
• Energy balance check
• Large number of experiments
• Graphing to identify discrepancy
• …..
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Regression analysis
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Single variable:
Minimum number of points depends on number of variable in the function (3 for the function above).Using the data we can set the system of equation to find the coefficients.
Lagrange interpolation
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Rewrite:
Find coefficients:
General form:
Model of complex system based on experimental data
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Example: chiller modelTOA
water
Building users (cooling coil in AHU)
TCWR=11oCTCWS=5oC
T Condensation
Chiller model
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EIRFPLCAPFTPP NOMINAL
OACWSOAOACWSCWS TTfTeTdTcTbaCAPTF 12
112
111
PLRcPLRbaEIRFPLR 333 NOMINALQ
QPLR
)(
Impact of temperatures:
Impact of capacity:
Two variable function fitting
What do you think
• Does correlation where R2=0.82 represent a good data modeling?
Mean:
Total sum of squares:
Sum of squares of residuals :
Coefficient of determination
Anscombe's quartet • Example of statistical misinterpretation of data
- all data have the same Mean (for x and y), Variance (for x and y)
- correlation R2: 0.816, linear regression: y=3.00+0.500·x
Anscombe's quartet • Example of statistical misinterpretation of data
- all curves have the same Mean (x, y), Variance (x, y)
- correlation R2: 0.816, linear regression
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Data set C
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Moral of the story
Francis Anscombe (in 1973) demonstrated • the importance of graphing data before
analyzing it • the effect of outliers on statistical properties
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